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Sunday, May 27, 2007

Riemann Rave

The most intriguing physical anecdote in du Sautoy'sThe Music of the Primes is the one about Keating's talk, entitled "Random matrix theory and some zeta-function moments", at the Vienna meeting in 1998.

Many mathematicians were dubious that the physicists could tell them anything beyond their observations on the statistics of Riemann zeroes and quantum chaos on surfaces. Yet another fine bottle of wine had been put forward, which Keating collected in Vienna after his talk. Nina Snaith and Keating had found a formula for generating the numbers known as zeta moments. The sequence begins 1,2,42,24024. The last number had only just been found by mathematicians, and yet it fell out of the physicists' formula when they looked at it just before Keating's talk. Links to the papers by Keating and Snaith, and also to Snaith's thesis, are available on Watkin's page.

Later, Keating went to the library in Goettingen to look up Riemann's original notes on both the zeta zeroes and hydrodynamics. He made the two requests to the librarian, but was handed only one pile of papers. Riemann had been working on both problems at the same time.

More recently, Carl and Michael have been making solid progress on understanding quantum black holes in M Theory. It would be really very interesting if the quantum chaos of such black holes had something to do with the Riemann zeroes. Oh, wait. It already does, because one uses 3x3 Hermitean matrices in a matrix theory setting.

Yes, Lubos discussed Vafa's paper back in 2005. The Calabi-Yau melting crystal geometry involves C^3 and blowups, which involves replacing a point with a complex projective space, e.g., CP^2 (see also hep-th/0309208. This is related to Kea's operad approach.

'The Max Plus approach in a few words':"... the so-called "Max Plus" algebra (essentially, the addition of two numbers consists of their maximum, whereas the multiplication of two numbers is the usual plus; for example 3 plus 5 equals 5, and 3 times 5 equals 8), or more generally, the dioid algebra (or idempotent semiring) ..."

http://cermics.enpc.fr/~cohen-g//SED/index-e.html

2 - Please also see comment #12 in 'Strings in Two Minutes or Less' of U-Duality.

It seems fully convincing that "proof" of RH (and other prime problems)are inseparable from First philosophy, ie. metaphysical problems.

In a way, it can be said that RH was allready "proven" by Gödels proof of death of the formalist dream, proof of formally unprovable mathematical fenomenological invariants aka "truths" - and hence the birth of proof theory, category theory etc.

I'm unable to fully follow Gödel's technical proof, but it appears (from Gödel numbers) that the "truth" of Gödels proof is codependant on the fenomenologically invariant "truth" of prime numbers and/or ordinality. Others hopefully understand it better and can correct my impression, if needed.

So, with the very convincing presuppositions that 1) RH is unprovable or "false" inside the axioms of set theory; 2) statistical distribution of primes is fenomenological invariant at least in this "anthropocentric" universe

then how does one (pun intended) proceed in formulating the metaphysical proof of RH?

The strategy that suggests itself is a version of proof by negation, diving blindly. If it is possible, by help of category theory, to "mindlessly as possible" to generate nonnatural sets of "N" with nonnatural distributions of primes, which however satisfy certain axiomatic requirements (set of mofphed axioms for arithmetics etc. that universes (with observers) can be build by) and if it then can be shown that these nonnatural sets of "N" (ie. with alternate distributions of primes) don't produce universes with similar phenomenological invariants (e.g. "laws" of physics) that are obvious to observers like us, perhaps this could be considered a metaphysical proof or RH (etc)?

Needles to say, methinks, such a task is a tall order and requires an utter madman or a bunch.

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